TQG is not that much a theory of quantum gravity, but a necessary
part of it. In particular, it provides a basic framework on which
to discuss within such a theory any aspect related to time, time evolution,
and duration. In this proposal, the part of quantum phase space that
deals with these aspects is modeled by an \(n-\) dimensional noncommutative
Torus, where each state on a NC-2-subTorus there corresponds to an
elementary and irreducible quantum process in a spacetime picture;
the NC-Torus arises here because for, say, three (of four) elements
that are causally related, one needs three generators that do not
commute among each other (this comes from the canonical quantization
of the Poisson brackets of phase space properties with weight functions
having causally connected supports), and this cannot be accomodated
just in the product of two NC-2-Torus (that model an individual quantum
elementary process), but one needs a missing third non-commutation
between these two, so to speak, and this can be done in the general
3 and \(4-\) NC-Torus. This type of phase space hasn't been discussed
in any of the current proposal for quantum gravity theories, despite
the fact that often some talk is informally given ([1]}, [2]})
about the behaviour of quantum time (like its hypothetical quantum
nature, with superpositions, etc.), that, really, can only be properly
justified if one has a model for this phase space.
